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Almost automorphy profile of solutions for difference equations of Volterra type

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Abstract

This work deals with the almost automorphic profile of solutions of the nonlinear Volterra difference equation \(u(n+1) = \lambda\sum_{j=-\infty}^{n}a(n-j)u(j) + f(n,u(n))\), n∈ℤ, for λ in a distinguished subset of the complex plane, where a(n) is a complex summable sequence and the perturbation f is a non-Lipschitz nonlinearity. Many illustrating remarks and examples are considered.

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Notes

  1. This seems reasonable and necessary since the uniform continuity condition is the main condition needed for the composition theorems of almost periodic functions, asymptotically almost periodic functions and pseudo-almost periodic functions (see [2, 32]).

  2. The convex hull of a set K is the set of all convex combinations of point in K: co(K):={θ 1 x 1+⋯+θ k x k :x i ∈K, θ i ≥0, i=1,…,k; θ 1+⋯+θ k =1}. As the name suggests, the convex hull co(K) is always convex. It is the smallest convex set that contain K.

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Acknowledgements

The results in this work were partially obtained during a visit (July-September 2012) of the second author to the Department of Mathematics and Statistics of Universidad de La Frontera under Programa Atracción e Inserción (PAI-MEC) Grant 80112008 (CONICYT-CHILE). He is grateful to professor Herme Soto and the Department of Mathematics and Statistics, for its generous hospitality and providing a stimulating atmosphere to work.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX, 78363-8202, USA

    Ravi P. Agarwal

  2. Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, 50540-740, Brazil

    Claudio Cuevas & Filipe Dantas

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  1. Ravi P. Agarwal
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  2. Claudio Cuevas
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  3. Filipe Dantas
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Correspondence to Claudio Cuevas.

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Agarwal, R.P., Cuevas, C. & Dantas, F. Almost automorphy profile of solutions for difference equations of Volterra type. J. Appl. Math. Comput. 42, 1–18 (2013). https://doi.org/10.1007/s12190-012-0615-3

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